Wave packet dynamics for a non-linear Schrodinger equation: Qualitative changes with changes in the initial width

TL;DR

This paper investigates how the initial width of a Gaussian wave packet influences its evolution in a nonlinear Schrödinger equation, revealing qualitative changes in propagation behavior depending on the initial conditions.

Contribution

It provides analytical predictions and numerical verification of how initial wave packet width affects dynamics in nonlinear Schrödinger systems, including free particles and harmonic oscillators.

Findings

Wave packets with initial width less than critical grow linearly in width.

Wave packets with initial width greater than critical become narrower and lose Gaussian shape.

Existence of a coupling strength for shape-preserving oscillations in harmonic oscillator case.

Abstract

The propagation of an initially Gaussian wave packet of width $\Delta_0$ in a cubic non-linear Schrodinger equation with a negative coupling constant for the nonlinear term is considered . It is predicted analytically and verified numerically that for a free particle if $\Delta_0$ is less than a critical value $\Delta_c$, then the packet will propagate in time with linearly growing width but for $\Delta>\Delta_c$, the packet will start becoming narrow and cease to be a Gaussian . For a simple harmonic oscillator, we find that for $\Delta_0$ smaller than a critical value, there always exist a coupling strength for which the packet simply oscillates about the mean position without changing its shape.